cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A103872 a(n) = 3*trinomial(n+1,0) - trinomial(n+2,0).

Original entry on oeis.org

0, 2, 2, 6, 12, 30, 72, 182, 464, 1206, 3170, 8426, 22596, 61074, 166194, 454950, 1251984, 3461574, 9611190, 26787378, 74916660, 210178458, 591347988, 1668172842, 4717282752, 13369522250, 37970114702, 108045430902
Offset: 0

Views

Author

Eric W. Weisstein, Feb 19 2005

Keywords

Comments

First differs from A059727 for n = 8.
Essentially twice A005043.

Links

Crossrefs

Cf. A005043, A059727.

Programs

  • Maple
    trinomial := n -> simplify(GegenbauerC(n,-n,-1/2)):
a := n -> 3*trinomial(n+1) - trinomial(n+2):
seq(a(n), n=0..27); # Peter Luschny, May 07 2016
  • Mathematica
    Table[(4*2^n (2n + 3)!! (3 Hypergeometric2F1[-2 - n, -1 - n, -3/2 - n, 1/4] - 4 Hypergeometric2F1[-2 - n, -2 - n, -3/2 - n, 1/4]))/(n + 2)!, {n, 0, 20}] (* Vladimir Reshetnikov, May 07 2016 *)

A305412 a(n) = F(n)*F(n+1) + F(n+2), where F = A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 3, 5, 11, 23, 53, 125, 307, 769, 1959, 5039, 13049, 33929, 88451, 230957, 603667, 1578823, 4130829, 10810469, 28295411, 74067401, 193893263, 507590495, 1328842801, 3478880593, 9107706243, 23844088085, 62424315227, 163428464759, 427860443429, 1120151837069
Offset: 0

Views

Author

Vincenzo Librandi, Jun 05 2018

Keywords

Links

Crossrefs

Cf. A001654, A269803.
Cf. A059769: F(n)*F(n+1) - F(n+2), with offset 3.
Equals A000045 + A286983.
First differences are listed in A059727 (after 0).

Programs

  • GAP
    List([0..35], n -> Fibonacci(n)*Fibonacci(n+1)+Fibonacci(n+2)); # Muniru A Asiru, Jun 06 2018
  • Magma
    [Fibonacci(n)*Fibonacci(n+1)+Fibonacci(n+2): n in [0..30]];
  • Mathematica
    Table[Fibonacci[n] Fibonacci[n+1] + Fibonacci[n+2], {n, 0, 30}]

Formula

G.f.: (1 - 5*x^2 - 2*x^3 + x^4)/((x + 1)*(1 - 3*x + x^2)*(1 - x - x^2)).
a(n) = 3*a(n-1) + a(n-2) - 5*a(n-3) - a(n-4) + a(n-5).
5*a(n) = (-1)^(n+1) +5*F(n+2) + A002878(n). - R. J. Mathar, Nov 14 2019

A011769 a(0) = 1, a(n+1) = 3 * a(n) - F(n)*(F(n) + 1), where F(n) = A000045(n) is n-th Fibonacci number.

Original entry on oeis.org

1, 3, 7, 19, 51, 141, 393, 1107, 3139, 8955, 25675, 73945, 213825, 620595, 1807263, 5279283, 15465139, 45420261, 133708777, 394446691, 1165855131, 3451793403, 10235554347, 30392965809, 90357645121, 268922897571, 801139867063, 2388683219347, 7127469430899
Offset: 0

Views

Author

N. J. A. Sloane, R. K. Guy

Keywords

References

  • L. Euler, (E326) Observationes analyticae, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 15, p. 59.
  • R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 575.
  • P. Henrici, Applied and Computational Complex Analysis. Wiley, NY, 3 vols., 1974-1986. (Vol. 1, p. 42.)
  • J. Riordan, Combinatorial Identities, Wiley, 1968, p. 74.
  • See also the references mentioned under A002426.

Links

Crossrefs

Cf. A002426.
Cf. A059727.

Programs

  • Haskell
    a011769 n = a011769_list !! n
a011769_list = 1 : zipWith (-) (map (* 3) a011769_list) a059727_list
-- Reinhard Zumkeller, Dec 17 2011
  • Maple
    A011769 := proc(n) if n = 0 then 1; else 3*procname(n-1)-combinat[fibonacci](n-1)*(1+combinat[fibonacci](n-1)) ; end if; end proc:
seq(A011769(n),n=0..40) ;
  • Mathematica
    nxt[{n_,a_}]:=Module[{fib=Fibonacci[n]},{n+1,3a-fib(fib+1)}]; Transpose[ [ nxt,{0,1},30]][[2]] (* or *) LinearRecurrence[{6,-8,-8,14,4,-3},{1,3,7,19,51,141},30] (* Harvey P. Dale, Jun 05 2015 *)

Formula

a(n) = +6*a(n-1) -8*a(n-2) -8*a(n-3) +14*a(n-4) +4*a(n-5) -3*a(n-6). [R. J. Mathar, Sep 04 2010]
G.f.: (1-3*x-3*x^2+9*x^3+3*x^4-3*x^5) / ( (3*x-1)*(1+x)*(x^2+x-1)*(x^2-3*x+1) ). - Sergei N. Gladkovskii, Dec 16 2011
a(n+1) = (1/10) * (3^n + 2*Lucas(2n) + Lucas(n) + (-1)^n ). - Ralf Stephan, Aug 10 2013
a(k) = 3^(k+1)*x^k/10 + (-1)^(k+1)*x^k/10 + p^(k+1)*x^k/5 + (-q)^(k+1)*x^k/5 + p^(2*k+2)*x^k/5 + q^(2*k+2)*x^k/5 ; p=(sqrt(5)+1)/2 , q=(sqrt(5)-1)/2 . - Sergei N. Gladkovskii, Dec 17 2011

Extensions

Values at n>=18 corrected by R. J. Mathar, Sep 04 2010

A354626 Numbers that can't be written as the sum of a Fibonacci number and the square of a Fibonacci number.

Original entry on oeis.org

15, 16, 18, 19, 20, 23, 24, 29, 31, 32, 36, 37, 39, 40, 41, 42, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 57, 58, 60, 61, 62, 63, 68, 70, 71, 73, 74, 75, 76, 78, 79, 81, 82, 83, 84, 86, 87, 88, 91, 92, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 115
Offset: 1

Views

Author

Angad Singh, Jul 09 2022

Keywords

Examples

			16 is a term since there does not exist any pair of integers i,j >= 0 such that Fibonacci(i) + Fibonacci(j)^2 = 16.

Crossrefs

Cf. A000045, A007598, A059727.

Formula

Numbers k such that the coefficient of x^k in the product (Sum_{i>=0} x^Fibonacci(i)) * (Sum_{j>=0} x^(Fibonacci(j)^2)) is 0.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.